On the Structural Constraint and Motion of 3-PRS Parallel Kinematic Machines

This paper presents a consistent analytic kinematic formulation of the 3-PRS parallel manipulator (PM) with a parasitic motion by embedding the velocity level structural constraint equation into the motion expression. Inverse rate kinematics (IRK) is solved with a simple constraint compatible velocity profile, which is obtained by projecting the instantaneous restriction space onto the motion space. Moreover, the systematic method to reveal the parasitic motion is introduced. Thus, the parasitic terms are automatically identified from the main motions. Unlike the usual approach, this study does not consider any explicit parasitic motion expression. Consequently, the derivation of constraint compatible input velocity, which comprises the parasitic term, is simplified. To incorporate the parasitic motion into the task velocity, constraint Jacobian of the manipulator is analytically obtained first. The manipulator Jacobian is extended to incorporate the passive joint’s information apart from the active joints and structural constraint. Hence, the dimension of the Jacobian matrix used to solve IRK is 9 × 6. The validity of the IRK is proved by the Bordered Gramian based forward rate kinematics (FRK). Then, an accurate numerical integration, RK4, is applied to the joint velocity of IRK to obtain the manipulator’s joint values. Consequently, the moving plate’s pose is obtained via forward position kinematics computed using integrated active and passive joint values for validation. The projection matrix used to get compatible constraint motion adjusts our input velocity and makes it compatible with the structural constraint policy, and the parasitic motion is embedded easily. Thus, an explicit formulation of the parasitic motion equation is not required, as the usual practice. Finally, the study presented numerical simulations to show the validity of the outlined resolutions. This paper’s result and analysis can be uniformly applied to other parallel manipulators with less than 6 DoFs.